Yes, using Cross-Origin XMLHttpRequest. Set the permissions in the manifest and use it like

var xhr = new XMLHttpRequest();
xhr.open("GET", "http://api.example.com/data.json", true);
xhr.onreadystatechange = function() {
  if (xhr.readyState == 4) {
    // WARNING! Might be injecting a malicious script!
    document.getElementById("resp").innerHTML = xhr.responseText;
    ...
  }
}
xhr.send();